p-group, metabelian, nilpotent (class 3), monomial
Aliases: C42.272D4, C42.400C23, C4.1062+ 1+4, C8⋊3D4⋊10C2, C4⋊D8⋊25C2, D4⋊D4⋊26C2, D4.D4⋊9C2, C4⋊C8.61C22, (C2×C8).62C23, C4⋊C4.153C23, (C2×C4).412C24, (C2×D8).71C22, C23.287(C2×D4), (C22×C4).501D4, C4⋊Q8.304C22, C4.104(C8⋊C22), C42.6C4⋊10C2, C8⋊C4.18C22, (C4×D4).106C22, (C2×D4).161C23, C22⋊C8.47C22, (C2×Q8).149C23, D4⋊C4.43C22, C4⋊1D4.165C22, C4⋊D4.192C22, (C2×C42).879C22, Q8⋊C4.44C22, (C2×SD16).32C22, C22.672(C22×D4), C2.57(D8⋊C22), (C22×C4).1083C23, C42.28C22⋊2C2, C22.26C24⋊20C2, C4.4D4.153C22, C2.83(C22.29C24), (C2×C4).541(C2×D4), C2.57(C2×C8⋊C22), (C2×C4○D4).174C22, SmallGroup(128,1946)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42.272D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1, cbc-1=dbd=a2b, dcd=b2c3 >
Subgroups: 492 in 217 conjugacy classes, 86 normal (28 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C2×C42, C4×D4, C4×D4, C4⋊D4, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C4⋊Q8, C2×D8, C2×SD16, C2×C4○D4, C2×C4○D4, C42.6C4, D4⋊D4, C4⋊D8, D4.D4, C42.28C22, C8⋊3D4, C22.26C24, C42.272D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8⋊C22, C22×D4, 2+ 1+4, C22.29C24, C2×C8⋊C22, D8⋊C22, C42.272D4
Character table of C42.272D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 8A | 8B | 8C | 8D | |
| size | 1 | 1 | 1 | 1 | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ10 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ13 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ14 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ15 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 2 |
| ρ17 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 2 | -2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ19 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | -2 | -2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ20 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | -2 | -2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ21 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ22 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from 2+ 1+4 |
| ρ23 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ24 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C8⋊C22 |
| ρ25 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
| ρ26 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 4i | 0 | 0 | 0 | -4i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from D8⋊C22 |
►On 64 points
Generators in S64
(1 30 59 15)(2 27 60 12)(3 32 61 9)(4 29 62 14)(5 26 63 11)(6 31 64 16)(7 28 57 13)(8 25 58 10)(17 39 54 43)(18 36 55 48)(19 33 56 45)(20 38 49 42)(21 35 50 47)(22 40 51 44)(23 37 52 41)(24 34 53 46)
(1 21 5 17)(2 51 6 55)(3 23 7 19)(4 53 8 49)(9 41 13 45)(10 38 14 34)(11 43 15 47)(12 40 16 36)(18 60 22 64)(20 62 24 58)(25 42 29 46)(26 39 30 35)(27 44 31 48)(28 33 32 37)(50 63 54 59)(52 57 56 61)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 34)(2 33)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 51)(10 50)(11 49)(12 56)(13 55)(14 54)(15 53)(16 52)(17 29)(18 28)(19 27)(20 26)(21 25)(22 32)(23 31)(24 30)(41 64)(42 63)(43 62)(44 61)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(64)| (1,30,59,15)(2,27,60,12)(3,32,61,9)(4,29,62,14)(5,26,63,11)(6,31,64,16)(7,28,57,13)(8,25,58,10)(17,39,54,43)(18,36,55,48)(19,33,56,45)(20,38,49,42)(21,35,50,47)(22,40,51,44)(23,37,52,41)(24,34,53,46), (1,21,5,17)(2,51,6,55)(3,23,7,19)(4,53,8,49)(9,41,13,45)(10,38,14,34)(11,43,15,47)(12,40,16,36)(18,60,22,64)(20,62,24,58)(25,42,29,46)(26,39,30,35)(27,44,31,48)(28,33,32,37)(50,63,54,59)(52,57,56,61), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,34)(2,33)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,29)(18,28)(19,27)(20,26)(21,25)(22,32)(23,31)(24,30)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,30,59,15)(2,27,60,12)(3,32,61,9)(4,29,62,14)(5,26,63,11)(6,31,64,16)(7,28,57,13)(8,25,58,10)(17,39,54,43)(18,36,55,48)(19,33,56,45)(20,38,49,42)(21,35,50,47)(22,40,51,44)(23,37,52,41)(24,34,53,46), (1,21,5,17)(2,51,6,55)(3,23,7,19)(4,53,8,49)(9,41,13,45)(10,38,14,34)(11,43,15,47)(12,40,16,36)(18,60,22,64)(20,62,24,58)(25,42,29,46)(26,39,30,35)(27,44,31,48)(28,33,32,37)(50,63,54,59)(52,57,56,61), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,34)(2,33)(3,40)(4,39)(5,38)(6,37)(7,36)(8,35)(9,51)(10,50)(11,49)(12,56)(13,55)(14,54)(15,53)(16,52)(17,29)(18,28)(19,27)(20,26)(21,25)(22,32)(23,31)(24,30)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([[(1,30,59,15),(2,27,60,12),(3,32,61,9),(4,29,62,14),(5,26,63,11),(6,31,64,16),(7,28,57,13),(8,25,58,10),(17,39,54,43),(18,36,55,48),(19,33,56,45),(20,38,49,42),(21,35,50,47),(22,40,51,44),(23,37,52,41),(24,34,53,46)], [(1,21,5,17),(2,51,6,55),(3,23,7,19),(4,53,8,49),(9,41,13,45),(10,38,14,34),(11,43,15,47),(12,40,16,36),(18,60,22,64),(20,62,24,58),(25,42,29,46),(26,39,30,35),(27,44,31,48),(28,33,32,37),(50,63,54,59),(52,57,56,61)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,34),(2,33),(3,40),(4,39),(5,38),(6,37),(7,36),(8,35),(9,51),(10,50),(11,49),(12,56),(13,55),(14,54),(15,53),(16,52),(17,29),(18,28),(19,27),(20,26),(21,25),(22,32),(23,31),(24,30),(41,64),(42,63),(43,62),(44,61),(45,60),(46,59),(47,58),(48,57)]])
Matrix representation of C42.272D4 ►in GL8(𝔽17)
| 0 | 0 | 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 | 0 | 0 |
| 13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 14 |
| 0 | 0 | 0 | 0 | 0 | 0 | 14 | 14 |
G:=sub<GL(8,GF(17))| [0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,3,14,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,14,3],[0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,3,14,0,0,0,0,0,0,14,14] >;
C42.272D4 in GAP, Magma, Sage, TeX
C_4^2._{272}D_4 % in TeX
G:=Group("C4^2.272D4"); // GroupNames label
G:=SmallGroup(128,1946);
// by ID
G=gap.SmallGroup(128,1946);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,891,675,248,4037,1027,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=b^2*c^3>;
// generators/relations
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